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FPKMC

First-Passage Kinetic Monte Carlo Methods for Reaction-Drift-Diffusion Processes

Within the cell, there is often a low concentration of certain types of proteins. This in turn causes finite number effects, such as spontaneous fluctuations in concentration. For many cellular processes the spatial distribution of chemical species plays an essential role. To address some of these challenges we present new computational methods, which allow for efficient particle-based simulations, combining ideas from Kinetic Monte Carlo Methods and First-Passage Time Analysis.

We do not necessarily expect the proteins to undergo Brownian motion as a consequence of interactions with cellular structures, material heterogeneity in the cytoplasm, and active transport. We introduce a computationally efficient method to simulate spatially dependent diffusion that is subjected to an arbitrary potential and reaction probabilities. Our method extends the First-Passage Kinetic Monte Carlo Methods (FPKMC) by adding drift and a spatially dependent diffusion. 

As a demonstration of this method, we look at a DNA-protein binding system where we take into account the geometry of the DNA and proteins that perform a combined one-dimensional and three-dimensional diffusion while searching for a binding site on the DNA. The protein undergoes one-dimensional diffusion along the DNA, which is modeled using our FPKMC method.

When the protein detaches from the DNA, we think of it as having two possible states: One state where it is unable to attach to the DNA and diffuses freely in three-dimensions, and another that takes place after the protein undergoes conformational change, which allows it to attach back to the DNA. This part is modeled using a three-dimensional diffusion with a sink term. The sink term represents the binding back to the DNA and is turned on after a period of time, which is represented using an exponential random variable. 

The task of how to numerically handle the sink term in the three-dimensional diffusion equation raises some interesting questions which align with the general theme of my work, that of coupling the discretization of the DNA to the discretization of the three-dimensional diffusion solver. We do this using the immersed boundary approach, by splitting the knot up into points where each one is represented by a kernel function that has good translational invariance properties. During the three-dimensional simulation we store the flux into the knot, which gives us the probability of reattaching to the DNA at a given location at a given time. This, in conjunction with the one-dimensional version of the FPKMC method, is used to obtain a complete model of one-dimensional and three-dimensional diffusion of a protein searching for a target site on a DNA.

We find that the geometry of the DNA plays a significant role in the reattachment location of the protein to the DNA. This work is carried out in collaboration with Ava J. Mauro, Justin Sharke, Samuel A. Isaacson, and Paul J. Atzberger.

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